212k views
1 vote
Suppose R is the triangle with vertices (−1,0),(0,1), and (1,0).

As an iterated integral, ∬ᵣ(5x+6y)²dA=∫Bₐ∫D (5x+6y)² dxdy with limits of integration

A =
B =
C =
D =

User Ziv Barber
by
7.6k points

1 Answer

4 votes

Final answer:

To evaluate the double integral ∫∫ℝ(5x+6y)²dA, we set the iterated integral's limits as A = 0, B = -1, C = 1 + x, and D = 1. We integrate first with respect to y, then with respect to x. The integration involves expanding the square of the function and integrating term by term.

Step-by-step explanation:

The question involves setting up and evaluating a double integral over a triangular region R with vertices at (-1,0), (0,1), and (1,0). The integrand is a function of x and y, specifically (5x+6y)². To solve this problem, it is necessary to determine the limits of integration for both the x and y integrals, taking into account the geometry of R.

First, we notice that the triangle is symmetrical about the y-axis, so the x-limits of integration will range from -1 to 1. The y-limits are a bit more involved. For a given x-value, y starts at 0 (at the base of the triangle) and goes up to the line connecting the points (-1, 0) and (0, 1), which has the equation y = 1 + x. So, our y-limits will be from 0 to 1 + x.

B and D are the limits for the outer integral (x-integral), so B = -1 and D = 1. A and C are the limits for the inner integral (y-integral), so A = 0 and C = 1 + x. Therefore, the iterated integral is:

∫-1∫1 ∫0∫1+x (5x+6y)² dy dx

To provide the correct answer by solving, you would integrate first with respect to y, from A to C, and then with respect to x, from B to D. The integration itself can be fairly complex, involving expanding the square and integrating term by term.

User Michael Mammoliti
by
8.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.